Is this a new thing, or has it always been this way? If its a new thing, is it related to the last time you changed the strings? Because every now and then you will get a bad string. Try changing the strings, and see if that is the issue.
If it has always seemed that way, I wouldn't jump right to the pickups; there could be a lot of culprits. You could have worn frets, or the neck could be out of whack. The bridges on those things suck, and that is a very possible perpetrator. And there is a solution for all of them, including the bridge. Bad frets or neck problems are a problem for your local reputable repair shop (hopefully you have at least one).
For the bridge, a lot of Jag and Jazzmaster players these days are switching to a
Mastery Bridge. It requires no modification to the guitar, so you can easily switch back if you ever sell the guitar, and it makes the guitar work a LOT better. They are being used by a lot of JM/Jag players, including Nils Cline and Jeff Tweedy of Wilco, Sonic Youth, Bill Frisell, and many others. The basic idea is the Mastery Bridge posts fit the thimbles tightly, so the bridge doesn't rock at all. This not only improves the tuning stability, but it will also help to transfer the vibration to the guitar more efficiently, which should improve the tone. They really are great bridges, though, and I highly recommend them.
(In the interest of full disclosure, the Mastery Bridge is made by one of my shop's employees - I have no financial stake in it, though, I simply think it may be helpful. We do keep them in stock at my shop, so I guess I could have a financial stake in selling you one, but you can order them directly through Mastery Bridge. They are frequently back ordered though, as the bridges are quite popular, and we almost always have them in stock due to our relationship with the company.)As for the electronics, if you DO decide to replace them, then the nature of the Jag and Jazzmaster makes it reasonable to remove the electronics and replace everything. For a 1966 Fender, the silliness of the vintage market means that re-soldering things will make a significant difference in the value of the guitar. That assumes that everything works, of course. If there is something broken, it is still better to fix it most of the time. If it were a`50's, that might be different, but I find that concept really...uh...silly. But if you are going to do it, you need to do everything, including the pickups, because if any of the solder joints are new the vintage market will go batty. I'm afraid I'm not real familiar with Jaguar pickups, but there are a lot of good ones out there these days.
Gabriel